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The Nijenhuis tensor of an almost complex manifold isIt measures the failure of the $(0,1)$ tangent distribution to be closed under the Lie bracket.

The restriction of $J$ to the $J$-invariant bundle $TY$ is an almost complex structure. Its Nijenhuis tensor is the restriction of the ambient Nijenhuis tensor because vector fields tangent to an embedded submanifold have tangent Lie bracket. The ambient tensor vanishes since $X$ is a complex manifold, so the Newlander-Nirenberg theorem makes the induced structure on $Y$ integrable. The inclusion has complex-linear differential and is therefore holomorphic; hence $Y$ is a complex submanifold.

For a complex submanifold, the holomorphic normal bundle isIf $Y$ is a smooth hypersurface, taking top exterior powers in the holomorphic conormal sequencegivesThe normal bundle of a hypersurface is $N_{Y/X}\cong {\mathcal{O}}_X(Y){|}_Y$, so the Adjunction formula is

On $X={\mathbb{CP}}^n\times {\mathbb{CP}}^m$, a bihomogeneous polynomial of bidegree $(d_1,d_2)$ is a section of the holomorphic line bundle $\mathcal{O}(d_1,d_2)$. Its zero locus is smooth precisely when the section is transverse to the zero section, equivalently when $F$ and all of its homogeneous first partial derivatives have no common projective zero. Sincethe Adjunction formula yieldsThus a smooth $F$ with $d_1=n+1$ and $d_2=m+1$ defines a complex submanifold with trivial canonical bundle.

Solved by gpt-5.6-sol high.

Complexifying the cotangent bundle of an almost complex manifold gives the $\pm i$ eigenspace decomposition $T^{\ast }X\otimes \mathbb{C}=T^{1,0\ast }X\oplus T^{0,1\ast }X$. DefineFor a form $\alpha$ of type $(p,q)$, the Dolbeault operator is the type-$(p,q+1)$ projection of $d\alpha$:

Solved by gpt-5.6-sol high.